Non-uniform sampling and spiral MRI reconstruction
John J. Benedettoa and Hui-Chuan Wub
a Department
b Digital
of Mathematics, University of Maryland, College Park, MD 20742
Systems Resources, Inc., 12450 Fair Lakes Circle, Suite 500,
Fairfax, VA 22033
ABSTRACT
There is a natural formulation of the Classical Uniform Sampling Theorem in the setting of Euclidean space, and in
the context of lattices, as sampling sets, and unit cells E, e.g., the Voronoi cell. For sampling at the Nyquist rate,
the sampling function corresponds to the sinc function, and it is an integral over E. The set E is a tile for Euclidean
space under translation by elements of the reciprocal lattice.
We have a constructive, implementable non-uniform sampling theorem in the context of uniformly discrete sampling sets and sets E, corresponding to the unit cells of the uniform sampling result. The set E has the property
that the translates by the sampling set of the polar set of E is a covering of Euclidean space. The theorem depends
on the theory of frames, and can be viewed as a modest generalization of a theorem of Beurling.
The application herein is to fast magnetic resonance imaging (MRI) by direct signal reconstruction from spectral
data on spirals.
Keywords: Non-uniform sampling, Fourier frames, Beurling density, MRI
1. INTRODUCTION
1.1. Background and Outline
d denote Rd when it is considered as the domain the Fourier
Let Rd be d-dimensional Euclidean space, and let R
transforms of signals defined on Rd . The Paley-Wiener space P WE is
d ) : supp ϕ∨ ⊆ E ,
P WE = ϕ ∈ L2 (R
d ) is the space of finite energy signals ϕ on R
d , i.e.,
where E ⊆ Rd is closed, L2 (R
ϕL2 (R
d ) =
ϕ∨ is the inverse Fourier transform of ϕ defined as
ϕ∨ (x) =
d
R
d
R
|ϕ(γ)|2 dγ
1/2
< ∞,
ϕ(γ)e2πi<x,γ> dγ,
and supp ϕ∨ denotes the support of ϕ∨ .
Beurling1 proved the following theorem for the case that E is the closed ball B(0, R) ⊆ Rd centered at 0 ∈ Rd
and with radius R.
d be uniformly discrete, and define
Theorem 1.1 (Beurling’s Theorem). Let Λ ⊆ R
ρ = ρ(Λ) = sup dist(ζ, Λ),
d
ζ∈R
where dist(ζ, Λ) is the Euclidean distance between the point ζ and the set Λ. If Rρ < 1/4, then Λ is a Fourier frame
for P WB(0,R) .
Recall that a set Λ is uniformly discrete if there is r > 0 such that
∀λ, γ ∈ Λ,
|λ − γ| ≥ r,
where |λ−γ| is the Euclidean distance between λ and γ. Fourier frames will be defined in Section 3, but a consequence
of their definition in the assertion of Beurling’s Theorem is that every finite energy signal f defined on E has the
representation
f (x) =
aλ (f )e2πi<x,λ>
(1)
λ∈Λ
in L2 -norm on E, where
λ∈Λ
|aλ (f )|2 < ∞. Beurling used the term “set of sampling” instead of “Fourier frame”.
We shall reformulate Beurling’s Theorem in Theorem 7.2, in terms of a covering condition; and we shall refer
to Theorem 7.2 as the Beurling Covering Theorem. Beurling’s Theorem stated above becomes an obvious corollary.
Our interest in this topic goes back to a problem about fast magnetic resonance imaging (MRI) posed to us by Dennis
Healy. Besides the present approach to non-uniform sampling using the power of Beurling’s results, there are other
approaches2 .3
In Section 2, we shall discuss the Classical Uniform Sampling Theorem for perspective with the result in Section 7
and for ultimately comparing lattice and tiling ideas with analogous notions from non-uniform sampling. Fourier
frames are introduced Section 3 in order to have a convenient structure in which to develop non-uniform sampling
formulas. Section 4 is devoted to a discussion of density criteria in the setting of completeness results, and such
criteria are essential for understanding effective signal reconstruction from non-uniformly spaced sampled values. We
state the fundamental characterization of Fourier frames in terms of density in Section 5.
The notion of balayage is defined in Section 6; and Beurling’s theorem (1959),4 describing the fundamental
relationship between balayage and Fourier frames, is stated. This relationship is an essential component of the
Beurling Covering Theorem in Section 7. We also take the opportunity to make some preliminary comments about
related on-going work relating dimensionality, tilings, lattices, and coverings.
In Section 8 we shall use the Beurling Covering Theorem to solve a mathematical version of the aforementioned
problem concerning fast MRI.5 Basically, spectral (Fourier transform) data of an unknown signal f is given on a
2 ; and the problem is to extract the original signal
discrete subset of finitely many interleaving spirals contained in R
2
2
f ∈ L (R ) from this data whenever possible. Our solution is contained in Examples 2 and 3. In these examples, we
2
shall provide a proof and algorithm for characterizing and constructing a uniformly discrete spectral subset Λ ⊆ R
on given interleaving spirals with the property that Λ is a Fourier frame in the sense of (1).
Complete proofs of the aforementioned results, along with an analysis and history of the various relevant concepts
of density, will appear in a forthcoming research tutorial,6 which also contains a complete bibliography.
1.2. Notation
We shall use standard notation from harmonic analysis7 .8
d ) is the convolution algebra of bounded Radon measures on R
d ; and, if Λ ⊆ R
d is closed, then
Further, M (R
d consisting of those elements supported by Λ. Integration over Euclidean
M (Λ) denotes the closed subspace of M (R)
space will be denoted by “ ”; and the Fourier transform f of f defined on Rd is formally given by
ϕ(γ) = f (γ) = f (x)e−2πi<x,γ> dx,
d . A(Rd ) is the space of absolutely convergent inverse Fourier transforms on Rd , and A (Rd ), its dual
where γ ∈ R
space when A(Rd ) is normed by
ϕ∨ A(Rd ) = ϕL1 (Rd ) ,
d for which
is the space of pseudo-measures on Rd . Bb (E) is the space of bounded continuous functions ϕ on R
∨
∨
supp ϕ ⊆ E, where E is closed and ϕ is the distributional inverse Fourier transform of ϕ. Clearly, Bb (E)∧ ⊆ A (Rd ).
d .
Finally, we write eλ (x) = e2πi<x,γ> for x ∈ Rd and γ ∈ R
2. CLASSICAL UNIFORM SAMPLING THEOREM
We begin by stating the Classical Uniform Sampling Theorem on R.
Theorem 2.1 (Classical Uniform Sampling Theorem).
Let T, Ω > 0 satisfy the condition that 0 < 2T Ω ≤ 1, and let s ∈ P W1/(2T ) satisfy the condition that s is a
bounded function on R which equals 1 on [−Ω, Ω]. Then
∀f ∈ P WΩ , f = T
f (nT )τnT s,
(2)
where the convergence of the sum is in L2 -norm and uniformly on R, and where (τnT s)(t) designates the translation
s(t − nT ).
It should be emphasized that there are non-bandlimited versions of Theorem 2.1, which can be interpreted as
modeling aliasing, including such decompositions in terms of Gabor and wavelet systems.2
The proof of Theorem 2.1 is elementary, and it depends essentially on periodization.7
requires the following definition.
The extension to Rd
Definition 2.2 (Lattices).
A lattice H ⊆ Rd is the image of Z d under some nonsingular linear transformation, i.e., H is a discrete subgroup
of Euclidean space Rd consisting of integral linear combinations of elements v1 , v2 , . . . , vd ∈ Rd , which form a basis
d of H is the lattice consisting of all γ ∈ R
d with the property that the inner
for Rd . The reciprocal lattice Λ ⊆ R
product x, γ is an integer nx for each x ∈ H.
d is a set U ⊆ R
d , not necessarily connected, such that
A unit cell or fundamental region of a lattice Λ ⊆ R
d
T = {γ + U : γ ∈ Λ} is a tiling or partition of R , i.e., the elements of T are pairwise disjoint and
d .
(γ + U ) = R
(3)
γ∈Λ
There are many possible choices for the unit cell of a given lattice. For example, the Voronoi cell or Brillouin zone
d closer to the origin than to any other lattice point.
is the unit cell of Λ defined as the set of all points in R
Theorem 2.3 (A d-dimensional Uniform Sampling Theorem).
d be a unit cell of the reciprocal lattice Λ. Define the sampling function
Let H ⊆ Rd be a lattice and let U ⊆ R
1
e2πix,γ dγ,
∀x ∈ Rd , sU (x) =
|U | U
where |U | is the Lebesgue measure of U , and let f ∈ L2 (Rd ) have the property that f = 0 a.e. off of U .
a. There is a continuous function fc on Rd such that f = fc a.e.
b. If f is continuous on Rd , then
f=
f (y)τy sU ,
y∈H
where the convergence of the sum is in L2 -norm and uniformly on Rd .
Note that Theorem 2.3, as a d-dimensional version of Theorem 2.1, is only stated for the sampling function sU
corresponding to the sinc function.
Remark 1 (Early Applications Motivating d-Dimensional Uniform Sampling).
a. In the mid-1950s, Brillouin (1956) discussed 3-dimensional uniform sampling with regard to some crystallographic problems (see the terminology ”Brillouin zone” before Theorem 2.3), and Bracewell (1956) used 2-dimensional
uniform sampling with regard to issues in radio astronomy. Miyakawa’s basic formulation (1959), in terms of
the “Nyquist relationships” between lattice and unit cell, was in the context of multivariate stochastic processes,
and Sasakawa (1960-61) provided applications of Miyakawa’s Theorem. Petersen and Middleton (1962) completed
Miyakawa’s approach, both theoretically and with many important examples. There is also Prosser’s sampling theorem and analysis of truncation error (1966). Although not “early” relative to the 1950s, we also mention Dubois’
application (1985) to video systems and Mersereau’s work (1979) on hexagonal sampling. In this discussion as well
in other parts of this paper, we have not provided specific references due to space considerations. We more than
compensate for these omissions in our research tutorial.6
b. Motivating-applications from mathematics go back to J.M. Whittaker’s uniform sampling theorem (1935) for
entire functions f of order less than two. The sampled values in Whittaker’s formula are f (m + in), m, n ∈ Z; and,
as noted by Pólya, the formula yielded a positive solution to Littlewood’s conjecture that if {f (m + in)} is bounded
then f is a constant. There are many other papers on d-dimensional uniform sampling, and Higgins’ exposition
(1985) is a reasonable place to start (but not to finish!).
c. The proofs of Theorem 2.3 are conceptually similar to those of Theorem 2.1. They depend essentially on
periodization in the guise of the proper Poisson Summation Formula (PSF) or of the canonical Fourier expansions
associated with Rd and its discrete subgroups. The Nyquist hypothesis 2T Ω ≤ 1 and bandlimited hypothesis
f ∈ P WΩ of Theorem 2.1 are replaced in Theorem 2.3 by the pairing H, Λ (where Λ is essential for choosing some
d ) and the hypothesis that f = 0 off of U , respectively.
unit cell U ⊆ R
d. In 1965 Igor Kluvánek proved Theorem 2.3 in the general setting of locally compact abelian groups. There
are applications of this result to saving bandwidth (even on R) and relations to the construction of single dyadic
orthonormal wavelets on Rd .6
3. FOURIER FRAMES
Definition 3.1 (Frames).
Let H be a separable Hilbert space with inner product x, y and norm x = x, x
Z } ⊆ H is a frame for H if there exist A, B > 0 such that
∀y ∈ H, Ay2 ≤
| y, xn |2 ≤ By2 .
1/2
d
. a. A sequence {xn : n ∈
(4)
A and B are frame bounds, and a frame is tight if A = B. A frame is exact if it is no longer a frame whenever any
one of its elements is removed.
The frame operator of the frame {xn } is the function S : H −→ H defined as Sy =
y, xn xn for all y ∈ H.
9
The theory of frames was developed by Duffin and Schaeffer.
b. An exact frame is a bounded unconditional basis and vice-versa. In particular, orthonormal bases (ONBs) are
exact frames and there are frames which are not exact frames.
An essential feature of frames {xn } ⊆ H is that they provide the harmonics for signal reconstruction formulas.
Frames may not be ONBs, but ONBs are not necessarily an advantage when it comes to noise reduction and stable
decompositions.
Theorem 3.2 (Frame Decomposition Theorem).
Let {xn : n ∈ Z d } ⊆ H be a frame for H with frame bounds A and B.
a. The frame operator S is a topological isomorphism with inverse S −1 : H −→ H. {S −1 xn } is a frame with
frame bounds B −1 and A−1 , and
y, xn S −1 xn in H.
(5)
∀y ∈ H, y =
y, S −1 xn xn =
b. If {xn } is a tight frame for H, if xn = 1 for all n, and if A = B = 1, then {xn } is an orthonormal basis for
H.
c. If {xn } is an exact frame for H, then {xn } and {S −1 xn } are biorthonormal, i.e.,
∀m, n,
xm , S −1 xn = δ(m, n) =
0 if m = n,
1 if m = n.
{S −1 xn } is the unique sequence in H which is biorthonormal to {xn }.
d. If {xn } is an exact frame for H, then the sequence resulting from the removal of any one element is not
complete in H, i.e., the linear span of the resulting sequence is not dense in H.
Definition 3.3 (Fourier Frames).
a. Let R > 0, and assume that the sequence {eλ : λ ∈ Λ} is a frame for H = L2 [−R, R]. This is clearly equivalent
to the assertion that there exist A, B > 0 such that
∀F ∈ P WR , AF 2 2 ≤
L (R )
λ∈Λ
|F (λ)|2 ≤ BF 2 2 .
L (R )
(6)
As such we say that {eλ : λ ∈ Λ} is a Fourier frame for L2 [−R, R], and by (5) we have
∀f ∈ L2 [−R, R], f =
aλ (f )eλ in L2 [−R, R].
(7)
λ∈Λ
(7) is a non-harmonic Fourier series, see Chapter VII of Paley and Wiener.10
b. The frame radius Rf (Λ) of Λ is
Rf (Λ) = sup{R ≥ 0 : {eλ } is a Fourier frame for L2 [−R, R]}.
d in Theorem 2.3. We let Λ ⊆ R
d be a lattice, and we let
Example 1. Let us switch the notation Rd and R
d
2 d
E ⊆ R be a unit cell of the reciprocal lattice. Then, if F ∈ L (R ) is a continuous function with the property that
f = F ∨ = 0a.e. off of E, we have the sampling formula
1
f (x) =
|E|
2πi<x,γ>
F (λ)e
1E (x)
(8)
λ∈Λ
d is a Fourier frame for P WE , then
in L2 (R). More generally, if Λ ⊆ R
∀f ∈ L2 (E),
f (x) =
cλ e2πi<x,γ>
λ∈Λ
2
in L (E), which can be compared with (8).
4. DENSITY CRITERIA FOR COMPLETENESS
d for which {eλ : λ ∈ Λ} is a Fourier frame for P WE for
In order to gain insight into the structure of sets Λ ⊆ R
d
To be
some E ⊆ R , it is reasonable to consider first criteria for which {eλ : λ ∈ Λ} is complete in the case Λ ⊆ R.
precise we let Λ = {λk : k ∈ Z, λ−1 < 0 ≤ λ0 , and limk→±∞ λk = ±∞} ⊆ R be a strictly increasing sequence, which
is uniformly discrete or separated in the sense that
∃r > 0 such that ∀k ∈ Z,
Further, for each R > 0, we let
λk+1 − λk ≥ r.
XR = span {eλ : λ ∈ Λ}
denote the closed linear span of {eλ : λ ∈ Λ} in L2 [−R, R]. Paley and Wiener10 (Chapter VI) refer to XR as the
“closure” of the set {eλ : λ ∈ Λ} of complex exponential functions. If XR = L2 [−R, R], then {eλ : λ ∈ Λ} is complete
in L2 [−R, R].
Definition 4.1 (The Closure of Sets of Complex Exponential Functions).
a. The radius of completeness Rc (Λ) of Λ is
Rc (Λ) = sup{R ≥ 0 : XR = L2 [−R, R]}.
Rc (Λ) is well-defined since it is clear that if R1 < R2 and XR2 = L2 [−R2 , R2 ], then XR1 = L2 [−R1 , R1 ].
b. An essential problem is to compute Rc (Λ), and in particular to find an intrinsic property of Λ so as to conclude
that XR = L2 [−R, R] for a given R. Note that XR = L2 [−R, R] for each R < Rc (Λ) and XR = L2 [−R, R] for each
R > Rc (Λ). Rc (Λ) is equal to the radii of completeness for the Lp -spaces Lp [−R, R], 1 ≤ p < ∞, as well as the space
C[−R, R] of continuous functions on [−R, R].
The following theorem is an early, fundamental, and deep result due to Paley and Wiener10 (Section 26).
Theorem 4.2 (A Paley-Wiener Completeness Theorem).
Assume that Λ has the property that λ0 = 0 and λ−k = −λk for k ≥ 1. For each γ > 0 let n(γ) be the cardinality
of {λk : k ≥ 1 and λk ≤ γ}. If
n(γ)
> 2R,
(9)
lim
γ→∞ γ
then XR = L2 [−R, R].
Remark 2 (Completeness and Density).
a. The “lim” on the left side of (9) is a density condition, and such conditions are essential hypotheses, not
only for completeness theorems such as Theorem 4.2 and Equation (15) below, but also for non-uniform sampling
formulas.
In engineering terms and in the context of non-uniform sampling, we can expect completeness if the number of
samples per unit time exceeds on average twice the largest frequency in the given signal, i.e., if the average sampling
rate exceeds the Nyquist rate. This sampling criteria is a density condition, and accurately quantifying the correct
density to obtain completeness is difficult.
Further, there are genuine engineering applications of some of these completeness theorems in uniquely determining signals from their non-uniformly spaced samples, e.g., Beutler’s work (1966) using results of Levinson.
b. Paley and Wiener’s book (1934)10 is the progenitor and driving force for an extensive and deep theory relating
refinements of Theorem 4.2 with various notions of density, see Definition 4.3. Some of the many notable works since
then are due to Levinson (1940), Duffin-Eachus (1942) and Duffin-Schaeffer9 (1952), Kahane (1962), Beurling and
Landau, e.g., Landau11 (1967), and Beurling and Malliavin (1962 and 1967).
There are also world class expositions due to Koosis (1970 and 1996) and Redheffer (1977) reflecting the authors’
profound understanding of the problems and their own seminal contributions from the 1960s, cf., Boas’ book (1954)
and a more recent and justly influential book due to Young (1980).
This material is expanded as background in a forthcoming tutorial on multidimensional non-uniform sampling.6
Definition 4.3 (Density Criteria).
be a strictly increasing uniformly discrete
Let Λ = {λk : k ∈ Z, λ−1 < 0 ≤ λ0 , and limk→±∞ λk = ±∞} ⊆ R
sequence, and define the function
nΛ =
k1[λk ,λk+1 )
whose distributional derivative is nΛ = λ∈Λ δλ . Clearly, if n : (0, ∞) → {0, 1, . . .} is defined by n(γ) = card{λk :
|λk | ≤ γ}, where “card” is cardinality, then n(γ) = nΛ (γ) − nΛ (−γ).
a. A reasonable definition of the density of Λ is
lim
γ→∞
n(γ)
2γ
when this limit exists. As such, and since nΛ (γk ) = k, we shall define the natural density of Λ as
Dn (Λ) = lim
|k|→∞
k
λk
when the limit in (10) exists. Otherwise, we consider the upper, resp., lower, natural densities
k
k
, resp., Dn− (Λ) = lim
.
|k|→∞ λk
|k|→∞ λk
Dn+ (Λ) = lim
b. Λ has uniform density Du (Λ) > 0 if
(10)
∃C > 0 such that ∀|γ| > 0, |nΛ (γ) − Du (Λ)γ| ≤ C.
(11)
Since nΛ (λk ) = k, the uniform density inequality (11) is equivalent to the condition that
∃C > 0 such that ∀k,
|
k
C
,
− Du (Λ)| ≤
λk
|λk |
or, equivalently,
|λk −
C
k
|≤
= L.
Du (Λ)
Du (Λ)
In particular, if Du (Λ) exists, then Dn (Λ) exists and equals Du (Λ). In fact, uniform density can be viewed as natural
density constrained by a convergence rate of 1/|λk |.
of length γ, let nI (γ) = card {λk ∈ I}. Define
c. For each γ > 0 and each interval I ⊆ R
n− (γ) = inf nI (γ)
I
and n+ (γ) = sup nI (γ).
I
The lower and upper Beurling densities of Λ are
n− (γ)
γ→∞
γ
Db− (Λ) = lim
n+ (γ)
,
γ→∞
γ
and Db+ (Λ) = lim
respectively. These limits exist since n− is superadditive and n+ is subadditive, although it is more common to
replace the limits by a lim and lim, respectively.
d. If Λ has uniform density Du (Λ) ∈ (0, ∞), then
Db− (Λ) = Db+ (Λ) = Du (Λ).
e. If Db− (Λ) = Db+ (Λ) = D(Λ), then the natural density Dn (Λ) exists and
Dn (Λ) = D(Λ).
Remark 3 (Beurling-Malliavin Density).
a. By definition of Rc (Λ), it is clear that Paley and Wiener’s Theorem 4.2 is equivalent to the assertion that
1 +
D (Λ) ≤ Rc (Λ).
(12)
2 n
Pólya (1929) introduced the notion that is called the Pólya maximum density Dp+ (Λ) of Λ, and it has the property
that Dp+ (Λ) ≥ Dn+ (Λ). In 1935 Levinson proved that if Λ is a positive sequence, then
1 +
D (Λ) ≤ Rc (Λ).
(13)
2 p
Further, Paley and Wiener10 (Theorem XXXIV on page 94) proved that if Λ = {λk : λ−k = −λk , k = 0, 1, . . .} has
uniform density Du (Λ) > 0 with small enough bound, then
1
Du (Λ).
2
Other upper bounds on Rc (Λ) are due to Koosis (1958) and Redheffer (1954).
Rc (Λ) ≤
(14)
b. In light of (12), (13), and (14), it is not unreasonable to conjecture that 12 Dn+ (Λ) = Rc (Λ) for symmetric
sequences Λ, e.g., Schwartz (1943). Kahane (1958) constructed a symmetric sequence with the properties that
Dn (Λ) = 0 and Rc (Λ) = ∞. Kahane’s sequence Λ is not uniformly discrete, cf., the results of Koosis (1960) and
Redheffer (1968).
c. In one of the highlights of 20th century analysis, Beurling and Malliavin (1962 and 1967) in 1960–1961 devised
a notion of density, denoted by Dbm (Λ), allowing them to prove
1
Dbm (Λ) = Rc (Λ).
2
(15)
The direction, Rc (Λ) ≥ 12 Dbm (Λ), is the easier to prove; and the direction, Rc (Λ) ≤ 12 Dbm (Λ), requires a deep study
of the canonical product,
z2
1− 2 ,
λk
using potential theory. The explanation par excellence is in Koosis’ book (1996).
5. CHARACTERIZATION OF FOURIER FRAMES
The following theorem is a characterization of Fourier frames in terms of density. Part a is due to Duffin and
Schaeffer9 (1952); part b is due to Landau11 (1967), although not using the term “frame”; and part c is due to
Jaffard12 (1991). Jaffard also characterized sets Λ of frequencies giving rise to Fourier frames as finite unions of
iniformly discrete sets at least one of which is uniformly dense. There is another deep characterization of such sets
by Ortega-Cerda and Seip (2000).
Theorem 5.1 (Fundamental Theorem of Fourier Frames).
a. If Λ has uniform density Du (Λ) > 2R, then {eλ : λ ∈ Λ} is a Fourier frame for L2 [−R, R].
b. If {eλ : λ ∈ Λ} is a Fourier frame for L2 [−R, R], then Db− (Λ) ≥ 2R.
c. If Rf (Λ) ∈ (0, ∞), then
Rf (Λ) =
1
sup{Du (Λ )},
2
where Λ ⊆ Λ has finite uniform density.
6. BALAYAGE
Definition 6.1. a. A convex, compact subset E ⊆ Rd is a symmetric body if it is symmetric about the origin in
the sense that if x ∈ E then −x ∈ E.
b. Let E ⊆ Rd be a symmetric body. Define
∀x ∈ Rd ,
xE = inf{r : x ∈ rE,
r ≥ 0},
where rE = {ry : y ∈ E}. It is elementary to prove that · E is a norm on Rd which is equivalent to the Euclidean
norm.
c. Let E ⊆ Rd be a symmetric body. The set
d : ∀x ∈ E,
E ∗ = {γ ∈ R
< x, γ >≤ 1},
is the polar set of E. It is clear that E ∗ is a symmetric body, and E ∗ is the unit sphere with respect to the norm
· E ∗ . In fact,
γE ∗ = sup | < x, γ > |
x∈E
and
xE = sup | < x, γ > |.
γ∈E ∗
d. Let z ∈ C d . Because of (16) it is natural to define zE as
zE = sup | < z, γ > |.
γ∈E ∗
(16)
Definition 6.2. Let E ⊆ Rd be a symmetric body with polar set E ∗ . An entire function ϕ on C d is of exponential
type E ∗ if
∀ > 0, ∃A > 0
such that
∀z ∈ C d ,
|ϕ(z)| ≤ A e2π(1+)
z
E
.
The classical Plancherel-Pólya Theorem, originally proved on R, has the following formulation on Rd , see pages
108–114 of Stein and Weiss.8
Theorem 6.3. (Plancherel-Pólya Theorem) Let E be a symmetric body and let p > 1. If ϕ is an entire function of
exponential type E ∗ , then
d ,
∀ξ ∈ R
1/p
|ϕ(γ + iξ)| dγ
p
d
R
≤ e2π
ξ
E∗
ϕLp (R
d ).
The Plancherel-Pólya Theorem can be used to prove the following Paley-Wiener Theorem. Generally, we shall
use the elementary sufficient conditions of the Paley-Wiener Theorem in order to obtain a function of exponential
type; and then we use this property of a given function in order to invoke Plancherel-Pólya.
d ). Then supp ϕ∨ ⊆ E
Theorem 6.4. (Paley-Wiener Theorem) Let E ⊆ Rd be a symmetric body, and let ϕ ∈ L2 (R
d of an entire function of exponential type E ∗ .
if and only if ϕ is the restriction to R
d ). The notation μ ∼E ν denotes the property that μ∨ = ν ∨
Definition 6.5. a. Let E ⊆ Rd , and let μ, ν ∈ M (R
on E.
d be a closed set. Balayage is possible for (E, Λ) if
b. Let E ⊆ Rd , and let Λ ⊆ R
d ), ∃ν ∈ M (Λ) such that μ∨ = ν ∨ on E,
∀μ ∈ M (R
d ), there is a ν ∈ M (Λ) such that μ ∼E ν.
i.e., for each μ ∈ M (R
The following is a consequence of the Open Mapping Theorem.
d be a closed set. Assume balayage is possible for (E, Λ). There is K > 0 such that
Let E ⊆ Rd , and let Λ ⊆ R
d ),
∀μ ∈ M (R
inf
ν∈M (Λ),μ∼E ν
νM (Λ) ≤ KμM (R
d ) .
(17)
The infimum of those K for which (17) holds is designated K(E, Λ).
Beurling considered the following two properties on a closed set E ⊆ Rd .
(α)
For each x0 ∈ E and each
> 0, there is μ ∈ M (B(x0 , ) ∩ E) such that lim|γ|→∞ μ
(γ) = 0;
(β) E is a set of spectral synthesis, i.e., T (f ) = 0 for all f ∈ A(Rd ) and all T ∈ A (Rd ) with the properties that
f = 0 on E and supp T ∈ A (Rd ).
Clearly, if int E = 0, then condition (α) is satisfied. There are other equivalent formulations of condition (β).
d ) with the
For example, condition (β) is satisfied if and only if ϕ(γ) dμ(γ) = 0 for all ϕ ∈ Bb (E) and all μ ∈ M (R
∨
d
property that μ = 0 on E. We shall use the fact that closed, convex sets E ⊆ R are sets of spectral synthesis.
The following result was proved by Beurling.4
Lemma 6.6. Let E ⊆ Rd be a compact set satisfying properties (α) and (β), and let E = {x : dist(x, E) ≤ }. If
K(E, Λ) < ∞, then there exists 0 such that
∀0 < <
0,
K(E , Λ) < ∞.
We shall also need the following result due to Ingham (1934). This lemma has major extensions and is intimately
related to the uncertainty principle in the context of Fourier analysis.
Lemma 6.7. Let Ω be a positive, increasing, continuous function on [0, ∞), and assume it satisfies the growth
conditions,
∞
Ω(γ)
dγ < ∞
γ2
0
and
d
R
For any
e−2Ω(|ξ|) dξ < ∞.
> 0, there exists an entire function h on C d with the properties that h(0) = 1,
d ,
∀ξ ∈ R
|h(ξ)| ≤ ce−Ω(|ξ|) ,
and supp h ⊆ B(0, ).
Using these results as well as Theorems 6.3 and 6.4, Beurling provided the ideas and methods for proving the
following theorem.
d be a closed set satisfying properties (α) and (β), assume E is symmetric with respect
Theorem 6.8. Let E ⊆ R
d be uniformly discrete. If balayage is possible for (E, Λ), then Λ is a Fourier frame for
to the origin, and let Λ ⊆ R
P WE .
7. BEURLING COVERING THEOREM
The following result was proved for the unit ball E = B(0, 1) ⊆ Rd by Beurling.1 As with Theorem 6.8, the proof
of the following result is a consequence of Beurling’s methods and ideas.6
d be points such that ξ −η ∈ {γ : γE ∗ ≤ 1/4}. There
Lemma 7.1. Let E ⊆ Rd be a symmetric body, and let ξ, η ∈ R
d ) with the properties that ν has support contained in the line passing through ξ
exists a discrete measure ν ∈ M (R
and η, ν({ξ}) = ν({η}) = 0, and
δξ ∼E cos(2πξ − ηE∗ )δη + ν.
This balayage of the Dirac measure is then used in conjunction with Theorem 6.8 to prove the following Beurling
Covering Theorem.
d be uniformly discrete set satisfying the covering
Theorem 7.2. Let E ⊆ Rd be a symmetric body, and let Λ ⊆ R
property
d .
τλ E ∗ = R
λ∈Λ
If r < 1/4, then Λ is a Fourier frame for P WrE .
The example at the end of Section 3 and Theorem 7.2 leads to the problem of relating coverings Λ + E ∗ and
tilings H + E when comparing the non-uniform and uniform sampling cases. Since tilings on Rd do not give rise
d for d > 4, when analyzing uniform sampling as a special case of Theorem 7.2, it is natural to
coverings on R
construct appropriate reciprocal sets H corresponding to uniformly discrete Λ so that the hypothesis in Theorem 7.2
generalizes tiling on Rd . This represents current work with Cabrelli and Molter.
8. A SPIRAL MRI RECONSTRUCTION ALGORITHM
As mentioned in the Introduction, we shall use the Beurling Covering Theorem to give a constructive irregular
sampling signal reconstruction method in the setting of interleaving spirals, see Example 3. Our method is much
more general than the geometry of interleaving spirals, and the particular examples which follow can in fact be
calculated using Beurling’s original formulation stated in the Introduction.
Example 2. Fix c > 0. We shall show how to choose a uniformly discrete subset ΛR of the spiral
2
Ac = {(cθ cos 2πθ, cθ sin 2πθ) : θ ≥ 0} ⊆ R
such that ΛR is a Fourier frame for P WB(0,R) , for some R > 0.
2 . There exists θ0 ∈ [0, 1) and r0 ≥ 0 such that (λ0 , γ0 ) = r0 (cos 2πθ0 , sin 2πθ0 ) ∈ R
2 . We first
i. Let (λ0 , γ0 ) ∈ R
observe that
dist((λ0 , γ0 ), Ac ) ≤ c/2,
(18)
and, in fact, sup(λ ,γ )∈R
2 dist((λ0 , γ0 ), Ac ) = c/2. To see this, note that either 0 ≤ r0 < cθ0 < c or that there is
0 0
n0 ∈ N ∪ {0} for which
c(n0 + θ0 ) ≤ r0 < c(n0 + 1 + θ0 ).
Thus,
dist((λ0 , γ0 ), Ac ) ≤ min{|c(n0 + θ0 ) − r0 |, |c(n0 + θ0 + 1) − r0 |}
since
|(λ0 , γ0 ) − (θ0 + n0 )e2πi(θ0 +n0 ) | = |r0 − c(θ0 + n0 )|.
Also,
c = c(n0 + θ0 + 1) − c(n0 + θ0 ) = c(n0 + θ0 + 1) − r0 + r0 − c(n0 + θ0 ) ≥ 2 min{|c(n0 + θ0 ) − r0 |, |c(n0 + θ0 + 1) − r0 |},
and (18) is obtained.
ii. Now, choose R for which Rc < 1/2, and then take δ > 0 such that (c/2 + δ)R < 1/4. Next, choose a uniformly
discrete set of points ΛR along the spiral Ac having curve distance between consecutive points less than 2δ, and
beginning within 2δ of the origin. Then the curve distance, and consequently the ordinary distance, from any point
2 to the spiral
on the spiral Ac to ΛR is less than δ. Further, as we showed in part i , the distance from any point in R
Ac is less than or equal to c/2. Thus, by the triangle inequality,
2 ,
∀ξ ∈ R
dist(ξ, ΛR ) ≤
c
+ δ = ρ.
2
Hence, Rρ < 1/4 by our choice of R and δ; and so we can invoke Beurling’s Covering Theorem, Theorem 7.2, or the
original formulation stated in the Introduction, to conclude that ΛR is a Fourier frame for P WB(0,R) .
M −1
Example 3. Given any R > 0 and c > 0. We shall show how to construct a finite interleaving set B = ∪k=1
Ak of
spirals
Ak = cθe2πi(θ−k/M ) : θ ≥ 0 , k = 0, 1, . . . , M − 1,
and a uniformly discrete set ΛR ⊆ B such that ΛR is a Fourier frame for P WB(0,R) . Thus, all of the elements of
L2 (B(0, R)) will have a decomposition in terms of the Fourier frame {eλ : λ ∈ ΛR }.
2 as ξ0 = r0 e2πiθ0 , where r0 ≥ 0
We begin by choosing M such that cR/M < 1/2. We shall write any given ξ0 ∈ R
and θ0 ∈ [0, 1). Then either 0 ≤ r0 < cθ0 < c or there is n0 ∈ N ∪ {0} for which
c(n0 + θ0 ) ≤ r0 < c(n0 + 1 + θ0 ).
In this latter case, we can find k ∈ {0, . . . , M − 1} such that
c(n0 + θ0 +
k
k+1
) ≤ r0 < c(n0 + θ0 +
).
M
M
Thus,
dist(ξ0 , B) ≤
c
.
2M
Next, we choose δ > 0 such that Rρ < 1/4, where ρ = (c/2M + δ).
Then, for each k = 0, 1, . . . , M − 1, we choose a uniformly discrete set Λk of points along the spiral Ak having
curve distance between consecutive points less than 2δ, and beginning within 2δ of the origin. The curve distance,
and consequently the ordinary distance, from any point on the spiral Ak to Λk is less than δ. Finally, we set
M −1
Λk . Thus, by the triangle inequality,
ΛR = ∪k=0
2 ,
∀ξ ∈ R
dist(ξ, ΛR ) ≤ dist(ξ, B) + dist(B, ΛR ) ≤
c
+ δ = ρ.
2M
Hence, Rρ < 1/4 by our choice of M and δ; and so we can invoke Beurling’s Covering Theorem, Theorem 7.2, or the
original formulation state in the Introduction, to conclude that ΛR is a Fourier frame for P WB(0,R) .
Note that since we are reconstructing signals on a space domain having area about R2 , we require essentially R
interleaving spirals. On the other hand, if we are allowed to choose the spiral(s) after we are given P WB(0,R) , then
we can choose ΛR contained in a single spiral Ac for c > 0 small enough.
REFERENCES
1. A. Beurling, “Local harmonic analysis with some applications to differential operators,” in Proc. Annual Science
Conference, pp. 109–125, Belfer Graduate School of Science, 1966.
2. J. J. Benedetto, “Irregular sampling and frames,” in Wavelets–A Tutorial in Theory and Applications, C. K.
Chui, ed., pp. 445–507, Academic Press, Boston, 1992.
3. K. Grochenig, “Non-uniform sampling in higher dimensions: from trigonometric polynomials to bandlimited
functions,” in Modern Sampling Theory: Mathematics and Applications, J. Benedetto and P. Ferreira, eds.,
p. Chapter 7, Birkhauser, Boston, 2000.
4. A. Beurling, “On balayage of measures in Fourier transforms (Seminar, Inst. for Advanced Studies, 1959–60,
unpublished),” in Collected Works of Arne Beurling, L. Carleson, P. Malliavin, J. Neuberger, and J. Wermer,
eds., Birkhauser, Boston, 1989.
5. M. Bourgeois, F. T. A. W. Wajer, D. van Ormondt, and D. Graveron-Demilly, “Reconstruction of MRI images
from non-uniform sampling, application to intrascan motion correction in functional MRI,” in Modern Sampling
Theory: Mathematics and Applications, J. Benedetto and P. Ferreira, eds., p. Chapter 16, Birkhauser, Boston,
2000.
6. J. Benedetto and H.-C. Wu, “The Beurling-Landau theory of Fourier frames and applications,” J. Fourier
Analysis and Applications 6, 2000.
7. J. J. Benedetto, Harmonic Analysis and Applications, CRC Press, Inc., Boca Raton, FL, 1997.
8. E. Stein and G. Weiss, An Introduction to Fourier Analysis on Euclidean Space, Princeton University Press,
Princeton, NJ, 1971.
9. R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic Fourier series,” Trans. Amer. Math. Soc. 72, pp. 341–
366, 1952.
10. R. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloquium Publ. 19,
New York, 1934.
11. H. J. Landau, “Necessary density conditions for sampling and interpolation of certain entire function,” Acta
Math. 117, pp. 37–52, 1967.
12. S. Jaffard, “A density criterion for frames of complex exponentials,” Michigan Math. J. 38, pp. 339–348, 1991.